Fast gradient-based methods with exponential rate

A hybrid control framework

Conference paper (2018)

Authors

Arman Sharifi K. Team Tamas Keviczky - Mechanical, Maritime and Materials Engineering
P. Mohajerin Esfahani Team Tamas Keviczky - Mechanical, Maritime and Materials Engineering
T. Keviczky Team Tamas Keviczky - Mechanical, Maritime and Materials Engineering
Research Group
Team Tamas Keviczky (Mechanical, Maritime and Materials Engineering) (TU Delft)
Copyright: © 2018 Arman Sharifi K., P. Mohajerin Esfahani, T. Keviczky
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Published Date

2018

Language

English

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Faculty

Mechanical, Maritime and Materials Engineering

Department

Delft Center for Systems and Control

Research Group

Team Tamas Keviczky

Abstract

Ordinary differential equations, and in general a dynamical system viewpoint, have seen a resurgence of interest in developing fast optimization methods, mainly thanks to the availability of well-established analysis tools. In this study, we pursue a similar objective and propose a class of hybrid control systems that adopts a 2nd-order differential equation as its continuous flow. A distinctive feature of the proposed differential equation in comparison with the existing literature is a state-dependent, time-invariant damping term that acts as a feedback control input. Given a user-defined scalar α, it is shown that the proposed control input steers the state trajectories to the global optimizer of a desired objective function with a guaranteed rate of convergence O(e−αt). Our framework requires that the objective function satisfies the so called Polyak–{Ł}ojasiewicz inequality. Furthermore, a discretization method is introduced such that the resulting discrete dynamical system possesses an exponential rate of convergence.